Solve {left(5sqrt{2}-3right)}^2-{left(3sqrt{2}+4right)}^2

$${ \left(5 \sqrt{ 2 } -3 \right) }^{ 2 } - { \left(3 \sqrt{ 2 } +4 \right) }^{ 2 }$$

Evaluate

$$25-54\sqrt{2}\approx -51.367532368$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5\sqrt{2}-3\right)^{2}.

$$25\left(\sqrt{2}\right)^{2}-30\sqrt{2}+9-\left(3\sqrt{2}+4\right)^{2}$$

The square of \sqrt{2} is 2.

$$25\times 2-30\sqrt{2}+9-\left(3\sqrt{2}+4\right)^{2}$$

Multiply 25 and 2 to get 50.

$$50-30\sqrt{2}+9-\left(3\sqrt{2}+4\right)^{2}$$

Add 50 and 9 to get 59.

$$59-30\sqrt{2}-\left(3\sqrt{2}+4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3\sqrt{2}+4\right)^{2}.

$$59-30\sqrt{2}-\left(9\left(\sqrt{2}\right)^{2}+24\sqrt{2}+16\right)$$

The square of \sqrt{2} is 2.

$$59-30\sqrt{2}-\left(9\times 2+24\sqrt{2}+16\right)$$

Multiply 9 and 2 to get 18.

$$59-30\sqrt{2}-\left(18+24\sqrt{2}+16\right)$$

Add 18 and 16 to get 34.

$$59-30\sqrt{2}-\left(34+24\sqrt{2}\right)$$

To find the opposite of 34+24\sqrt{2}, find the opposite of each term.

$$59-30\sqrt{2}-34-24\sqrt{2}$$

Subtract 34 from 59 to get 25.

$$25-30\sqrt{2}-24\sqrt{2}$$

Combine -30\sqrt{2} and -24\sqrt{2} to get -54\sqrt{2}.

$$25-54\sqrt{2}$$

Expand

$$25-54\sqrt{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5\sqrt{2}-3\right)^{2}.

$$25\left(\sqrt{2}\right)^{2}-30\sqrt{2}+9-\left(3\sqrt{2}+4\right)^{2}$$

The square of \sqrt{2} is 2.

$$25\times 2-30\sqrt{2}+9-\left(3\sqrt{2}+4\right)^{2}$$

Multiply 25 and 2 to get 50.

$$50-30\sqrt{2}+9-\left(3\sqrt{2}+4\right)^{2}$$

Add 50 and 9 to get 59.

$$59-30\sqrt{2}-\left(3\sqrt{2}+4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3\sqrt{2}+4\right)^{2}.

$$59-30\sqrt{2}-\left(9\left(\sqrt{2}\right)^{2}+24\sqrt{2}+16\right)$$

The square of \sqrt{2} is 2.

$$59-30\sqrt{2}-\left(9\times 2+24\sqrt{2}+16\right)$$

Multiply 9 and 2 to get 18.

$$59-30\sqrt{2}-\left(18+24\sqrt{2}+16\right)$$

Add 18 and 16 to get 34.

$$59-30\sqrt{2}-\left(34+24\sqrt{2}\right)$$

To find the opposite of 34+24\sqrt{2}, find the opposite of each term.

$$59-30\sqrt{2}-34-24\sqrt{2}$$

Subtract 34 from 59 to get 25.

$$25-30\sqrt{2}-24\sqrt{2}$$

Combine -30\sqrt{2} and -24\sqrt{2} to get -54\sqrt{2}.

$$25-54\sqrt{2}$$